Water based does not have to mean that the resulting paint, when dried and cured, is water soluble. Acrylics are usually water based, but if you get any in the wrong place, they need a lot more than water to shift after they have cured.
Just thinking out loud, but if a stain was thinned with something runnier than water, like isopropyl, would it soak in better?
I still prefer oil based paint, I can usualy predict what my results will be and the surfce rarely needs any extra protection.